In the xy-plane , line l intersects a circle with center

ALL LINES WHICH ARE PERPENDICULAR TO TANGENT WILL PASS THROUGH CENTRE.

IMO C.

SEE DIAG FOR EXPLANATION.

In the xy-plane , line l intersects a circle with center at origin.Is the slope of line l equal to zero?

1) line l passes through second quadrant.
2) line l is perpendicular to the tangent of the circle.

FS1: Line l passing through second quadrant and intersecting the circle can be either parall to x-axis(in which case its slope is 0) or not parallel to x-axis(slope not eual to 0). Hence insufficient.
FS2: Line which is perpendicular to the tangent must pass thought centre (since angle betwwen radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). Not sufficient.

Combining the above two, the only possibility is when the slope is negatice but not zero (x-axis is not in II quadrant). Hence slope iof line l is not zero for any case.
Sufficient. So answer should be C.

The 2nd statement does not imply that the line l has to pass through the tangent point. It could be any parallel line perpendicular to the tangent.
Shouldn't the answer be E?

Please clarify??

You are right the answer must be E:
_________________

Bunuel,

Thank you so much for the quick response. You are the best :D

Stmt1: If any line l passes through second quadrant and intersects the circle, then it can be
- either parallel to x-axis => slope is 0
- or not parallel to x-axis => slope not equal to 0.
Hence insufficient.

Stmt2: If a line is perpendicular to the tangent must pass thought centre (since angle between radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). SO this is also Not sufficient.

If we combine two stmts, the only possibility left is negative slope but not zero (x-axis is not in II quadrant). Hence slope of line will never be 0.
Sufficient. Ans- C.

Target question: Is the slope of line l equal to zero?

IMPORTANT: Since a circle can have an infinite number of tangent lines, the answer here is E.
To prove this, let's jump straight to . . .

Statements 1 and 2 combined
Here are two possible scenarios that meet the two given conditions:
Case a:


Case b:


In once case, the slope of line l equals 0, and in the other case, the slope of line l does not equal 0.
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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